This is where tropical geometry can help. First, we find a graph of of genus \(g\)  that satisfies the statement of the Brill–Noether theorem. Namely, a graph \(\Gamma \)  such that \(\dim (W^r_d(\Gamma ))=\rho \) . Next, we observe that \(\Gamma \)  is the tropicalization of some algebraic curve \(C\)  [1]}.
Now, the Specialization Lemma REF  implies that \(C\)  satisfies the statement of Brill–Noether as well. While we don't have an explicit description of this curve \(C\) , we now know that it exists!
